Hierarchies in directed networks
Nikolaj Tatti
HIIT, Aalto University, Espoo, Finland, [email protected]
Abstract—Interactions in many real-world phenomena can
be explained by a strong hierarchical structure. Typically, this
structure or ranking is not known; instead we only have observed
outcomes of the interactions, and the goal is to infer the hierarchy
from these observations. Discovering a hierarchy in the context of
directed networks can be formulated as follows: given a graph,
partition vertices into levels such that, ideally, there are only
edges from upper levels to lower levels. The ideal case can only
happen if the graph is acyclic. Consequently, in practice we have
to introduce a penalty function that penalizes edges violating
the hierarchy. A practical variant for such penalty is agony,
where each violating edge is penalized based on the severity
of the violation. Hierarchy minimizing agony can be discovered
in O(m2 ) time, and much faster in practice. In this paper we
introduce several extensions to agony. We extend the definition for
weighted graphs and allow a cardinality constraint that limits the
number of levels. While, these are conceptually trivial extensions,
current algorithms cannot handle them, nor they can be easily
extended. We provide an exact algorithm of O(m2 log n) time by
showing the connection of agony to the capacitated circulation
problem. We also show that this bound is in fact pessimistic and
we can compute agony for large datasets. In addition, we show
that we can compute agony in polynomial time for any convex
penalty, and, to complete the picture, we show that minimizing
hierarchy with any concave penalty is an NP-hard problem.
I. I NTRODUCTION
Interactions in many real-world phenomena can be explained by a strong hierarchical structure. As an example, it
is more likely that a line manager in a large, conservative
company will write emails to her employees than the other
way around. Typically, this structure or ranking is not known;
instead we only have observed outcomes of the interactions,
and the goal is to infer the hierarchy from these observations.
Discovering hierarchies or ranking has applications in various
domains, such as, ranking players [3], discovering dominant
animals [8], hierarchy discovery in social networks [11], and
summarizing browsing behaviour [10].
We consider the following problem of discovering hierarchy
in the context of directed networks: given a directed graph,
partition vertices into ranked groups such that there are only
edges from upper groups to lower groups.
Unfortunately, such a partitioning is only possible when
the input graph has no cycles. Consequently, a more useful
problem definition is to define a penalty function p on the
edges. This function should penalize edges that are violating
a hierarchy. Given a penalty function, we are then asked to
find the hierarchy that minimizes the total penalty.
The feasibility of the optimization problem depends drastically on the choice of the penalty function. If we attach a
constant penalty to any edge that violates the hierarchy, that
is, the target vertex is ranked higher or equal than the source
vertex, then this problem corresponds to a feedback arc set
problem, a well-known NP-hard problem [1], even without a
known constant-time approximation algorithm [4].
A more practical variant is to penalize the violating edges
by the severity of their violation. That is, given an edge
(u, v) we compare the ranks of the vertices r(u) and r(v)
and assign a penalty of max(r(u) − r(v) + 1, 0). Here, the
edges that respect the hierarchy receive a penalty of 0, edges
that are in the same group receive a penalty of 1, and penalty
increases linearly as the violation becomes more severe. This
particular score is referred as agony. Minimizing agony was
introduced by Gupte et al. [6] where the authors provide an
exact O(nm2 ) algorithm, where n is the number of vertices
and m is the number of edges. A faster discovery algorithm
with the computational complexity of O(m2 ) was introduced
by Tatti [17]. In practice, the bound O(m2 ) is very pessimistic
and we can compute agony for large graphs in reasonable time.
In this paper we focus on agony, and provide the following
main extensions for discovering hierarchies in graphs.
weighted graphs: We extend the notion of the agony to
graphs with weighted edges. Despite being a conceptually
trivial extension, current algorithms [6, 17] for computing
agony are specifically design to work with unit weights, and
cannot be used directly or extended trivially. Consequently, we
need a new approach to minimize the agony, and in order to
do so, we demonstrate that we can transform the problem into
a capacitated circulation, a classic graph task known to have
a polynomial-time algorithm.
cardinality constraint: The original definition of agony
does not restrict the number of groups in the resulting partition.
Here, we introduce a cardinality constraint k and we are asking
to find the optimal hierarchy with at most k groups. This
constraint works both with weighted and non-weighted graphs.
Current algorithms for solving agony cannot handle cardinality
constraints. Luckily, we can enforce the constraint when we
transform the problem into a capacitated circulation problem.
convex penalties: Minimizing agony uses linear penalty for
edges. We show that if we replace the linear penalty with a
convex penalty, we can still solve the problem in polynomial
time. However, this extension increases the computational
complexity.
concave penalties: To complete the picture, we also study
concave edge penalties. We show that in this case discovering
the optimal hierarchy is an NP-hard problem. This provides a
stark difference between concave and convex penalties.
The proofs are in the complete version of the paper.1
1 http://research.ics.aalto.fi/dmg/
II. P RELIMINARIES AND PROBLEM DEFINITION
We begin with establishing preliminary notation and then
defining the main problem.
The main input to our problem is a weighted directed graph
which we will denote by G = (V, E, w), where w is a function
mapping an edge to real positive number. If w is not provided
we assume that each edge has a weight of 1. We will often
denote n = |V | and m = |E|.
As mentioned in the introduction, our goal is to partition
vertices V . We express this partition with a rank assignment
r, a function mapping a vertex to an integer. To obtain the
groups from the rank assignment we simply group the vertices
having the same rank.
Given a graph G = (V, E) and a rank assignment r, we
will say that an edge (u, v) is forward if r(u) < r(v),
otherwise edge is backward, even if r(u) = r(v). Ideally,
rank assignment r should not have backward edges, that is,
for any (u, v) ∈ E we should have r(u) < r(v). However, this
is only possible when G is a DAG. For a more general case,
we assume that we are given a penalty function p, mapping an
integer to a real number. The penalty for a single edge (u, v)
is then equal to p(d), where d = r(u) − r(v). If p(d) = 0,
whenever d < 0, then the forward edges will receive 0 penalty.
We highlight two penalty functions. The first one assigns a
constant penalty to each backward edge, the second penalty
function assigns a linear penalty to each backward edge,
(
1 if d ≥ 0
pc (d) =
pl (d) = max(0, d + 1) .
0 otherwise ,
For example, an edge (u, v) for which r(u) = r(v) is
penalized by pl (r(u) − r(v)) = 1, the penalty is equal to 2 if
r(u) = r(v) + 1, and so on.
Given a penalty function and a rank assignment we define
the the score to be the sum of the weighted penalties.
Definition 1. Assume a weighted directed graph G =
(V, E, w) and a rank assignment r. Assume also a cost
function p mapping an integer to a real number. We define
a score for a rank assignment to be
X
q(G, r, p) =
w(u, v)p(r(u) − r(v)) .
(u,v)∈E
We will refer the score q(G, r, pl ) as agony.
Example 2. Consider the left ranking r1 of a graph G given
in Figure 1. This ranking has 5 backward edges: the penalty is
q(G, r1 , pc ) = 5. On the other hand, there are with 2 edges,
(i, a) and (e, g), with the agony of 1, 2 edges has agony of 2,
and (d, b) has agony of 3. Hence, agony is equal to
q(G, r1 , pl ) = 2 × 1 + 2 × 2 + 1 × 3 = 10 .
The agony for the right ranking r2 is q(G, r2 , pl ) = 7.
Consequently, r2 yields a better ranking in terms of agony.
i
a
f
i
g
e
b
c
b
c
h
d
d
g
f
e
h
Figure 1. Toy graphs. Backward edges are represented by dotted lines, while
the forward edges are represented by solid lines. Ranks are represented by
dashed grey horizontal lines.
Problem 1. Given a graph G = (V, E, w), a cost function p,
and an integer k, find a rank assignment r minimizing q(r, G)
such that 0 ≤ r(v) ≤ k − 1 for every v ∈ V . We will denote
the optimal score by q(G, k, p).
We should point out that we have an additional constraint by
demanding that the rank assignment may have only k distinct
values, that is, we want to find at most k groups. Note that
if we assume that the penalty function is non-decreasing and
does not penalize the forward edges, then setting k = |V | is
equivalent of ignoring the constraint. This is the case since
there are at most |V | groups and we can safely assume that
these groups obtain consecutive ranks. However, an optimal
solution may have less than k groups, for example, if G has no
edges and we use pl (or pc ), then a rank assigning each vertex
to 0 yields the optimal score of 0. We should also point out that
unlike with pc , it is possible that the ranking minimizing agony
must yield a partition which contains non-singleton groups.
It is easy to see that minimizing q(G, pc ) is equivalent to
finding a directed acyclic subgraph with as many edges as possible. This is known as F EEDBACK A RC S ET (FAS) problem,
which is NP-complete [1]. On the other hand, if we assume
that G has unit weights, and set k = |V |, then minimizing
agony has a polynomial-time O(m2 ) algorithm [6, 17].
III. C OMPUTING AGONY
In this section we present a technique for minimizing agony,
that is, solving Problem 1 using pl as a penalty. In order to
do this we show that this problem is in fact a dual problem
of the known graph problem, closely related to the minimum
cost max-flow problem.
A. Agony is a dual problem of Circulation
Minimizing agony is closely related to a circulation problem, where the goal is to find a circulation with a minimal
cost satisfying certain balance equations.
Problem 2 (C IRCULATION). Assume a directed graph G =
(V, E, s, c) with weights s and capacities c on edges. Find a
flow f such that 0 ≤ f (e) ≤ c(e) for every e ∈ E and
X
X
f (v, u) =
f (u, v), for every v ∈ V
(v,u)∈E
(u,v)∈E
minimizing
X
s(u, v)f (u, v) .
(u,v)∈E
We can now state our main optimization problem.
a
We denote the above sum as circ(G).
α
)
c
d
H (G, 4) ω
IV. A LTERNATIVE PENALTY FUNCTIONS
This problem is known as capacitated circulation problem,
and can be solved in O(m log n(m + n log n)) time with an
algorithm presented by Orlin [13]. We should stress that we
allow s to be negative, otherwise the optimal solution would
be a zero flow. We also allow capacities for certain edges to
be infinite, that is, f (e) ≤ c(e) is not enforced, if c(e) = ∞.
In order to transform the problem of minimizing agony to
the capacitated circulation problem, assume that we are given a
graph G = (V, E, w) and an integer k. We define a graph H =
(W, F, s, c) as follows. The vertex set W consists of 2 groups:
(i) |V | vertices, each vertex corresponding to a vertex in G (ii)
2 additional vertices α and ω. For each edge e = (u, v) ∈ E,
we add an edge f = (u, v) to F . We set c(f ) = w(e) and
s(e) = −1. We add edges (v, ω) and (α, v) for every v ∈ V
with a weight of 0 and a capacity of ∞, and finally we add
(ω, α) with s(ω, α) = k − 1 and c(ω, α) = ∞. We will denote
this graph by H (G, k) = H.
Example 3. Consider G = (V, E), a graph with 4 vertices and
4 edges, given in Figure 2. Set cardinality constraint k = 4.
In order to construct H (G, k) we add two additional vertices
α and ω to enforce the cardinality constraint k. We set edge
costs to −1 and edges capacities to be the weights of the input
graph. We connect α and ω with a, b, c, and d, and finally we
connect ω to α. The resulting graph is given in Figure 2.
The following proposition shows the connection between
the agony and the capacitated circulation problem.
Proposition 4. Assume a weighted directed graph G =
(V, E, w) and an integer k. Then
q(G, k, pl ) = −circ(H (G, k))
.
We have shown that we can find ranking minimizing edge
penalties pl in polynomial time. In this section we consider
alternative penalties. More specifically, we consider convex
penalties which are solvable in polynomial time, and concave
penalties which are NP-complete.
A. Convex penalty function
We say that the penalty function is convex if p(x) ≤
(p(x − 1) + p(x + 1))/2 for every x ∈ Z.
Let us consider a penalty function that can be written as
ps (x) =
`
X
max(0, αi (x − βi )),
i=1
where αi > 0 and βi ∈ Z for 1 ≤ i ≤ `. This penalty
function is convex. On the other hand, if we are given a convex
penalty function p such that p(x) = 0 for x < 0, then we can
safely assume that an optimal rank assignment will have values
between 0 and |V | − 1. We can define a penalty function ps
with ` ≤ |V | terms such that ps (x) = p(x) for x < |V |.
Consequently, finding an optimal rank assignment using ps
will also yield an optimal rank assignment with respect to p.
Note that pl is a special case of ps . This hints that we can
solve q(G, k, ps ) with a technique similar to the one given in
Section III. In order to do this, assume a graph G = (V, E, w)
and an integer k. Set n = |V | and m = |E|. We define a
graph H = (W, F, c, b) as follows. The vertex set W consists
of 2 groups: (i) n vertices, each vertex corresponding to a
vertex in G (ii) 2 additional vertices α and ω. For each edge
e = (v, w) ∈ E, we add ` edges fi = (u, v) to F . We set
s(fi ) = βi and c(fi ) = αi w(e). We add edges (v, ω) and
(ω, v) for every v ∈ V with 0 weight and infinite capacity.
Finally we add (ω, α) with c(t, s) = k − 1. We denote this
graph by H (G, k, ps ) = H.
B. Algorithm for minimizing agony
a
1
2
−1(1)
6
a
b
−
1
2)
1(
G
c
H (G, 3, ps )
b3
3(2)
4)
3(
Proposition 4 states that we can compute agony but it
does not provide direct means to discover an optimal rank
assignment. However, a closer look at the proof reveals that
minimizing agony is a dual problem of C IRCULATION.
Luckily, the algorithms for solving C IRCULATION by Edmonds and Karp [2] or by Orlin [13] solve the dual problem,
and are guaranteed to have integer-valued solution as long as
the capacities s(u, v) are integers, which is the case for us.
If we are not enforcing the cardinality constraint, that is,
we are solving q(G, k) with k = |V |, we can obtain a
1)
Figure 2. Toy graph G and the related circulation graph H (G, 4). Edge costs
and capacities for (α, v) and (v, ω) are omitted to avoid clutter.
2)
d
1(
c
3(
G
Proposition 5. Assume a graph G, and set k = |V |. Let {Ci }
be the strongly connected components of G, ordered in a topological order.
Let ri be the ranking minimizing q(G(Ci ), |Ci |).
Pi−1
Let bi =
j=1 |Cj |. Then the ranking r(v) = ri (v) + bi ,
where Ci is the component containing v, yields the optimal
score q(G, k).
−
1)
3(∞)
2)
(2
1
b
(
−1
1
−1
2
−1(1)
a
b
1
−1
(
a
significant speed-up by decomposing G to strongly connected
components, and solve ranking for individual components.
c
3
Figure 3. Toy graph G and the related circulation graph H (G, 3, ps ). To
avoid clutter the vertices α and ω and the adjacent edges are omitted.
Example 6. Consider a graph G given in Figure 3 and a
penalty function ps (d) = max(0, d+1)+2 max(0, d−3). The
graph H = H (G, 3, ps ) has 5 vertices, the original vertices
and the two additional vertices. Each edge in G results in two
edges in H. This gives us 6 edges plus the 7 edges adjacent
to α or ω. The graph H without α and ω is given in Figure 3.
Proposition 7. Assume a weighted directed graph G =
(V, E, w), an integer k, and a penalty ps cost. Then
q(G, k, ps ) = −circ(H (G, k, ps ))
.
Note that this proposition is a generalization of Proposition 4. If we set ` = 1, α1 = 1, and β1 = −1, then
H (G, k, ps ) = H (G, k), and Proposition 7 is equivalent to
Proposition 4. The proof of Proposition 7 is essentially equivalent to the proof of Proposition 4, and we omit the proof for
brevity. More importantly, we can discover the optimal ranking
by solving the capacitated circulation problem, similarly as we
did when minimizing agony.
Finally, let us address the computational complexity of the
problem. The circulation graph H (G, k, ps ) will have n + 2
vertices and `m + n. If the penalty function p is convex, then
we need at most ` = n functions to represent p between the
range of [0, n − 1]. Moreover, if we enforce the cardinality
constraint k, we need only ` = k components. Consequently,
we will have at most dm + n, edges where d = min(k, `, n)
for ps , and d = min(k, n) for a convex penalty p. This gives
us computational time of O(dm log n(dm + n log n)).
B. Concave penalty function
We have shown that we can solve Problem 1 for any
convex penalty. Let us consider concave penalties, that is
penalties for which p(x) ≥ (p(x − 1) + p(x + 1))/2. There
is a stark difference compared to the convex penalties as the
minimization problem becomes computationally intractable.
Proposition 8. Assume a monotonic penalty function p : Z →
R such that p(x) = 0 for x < 0, p(2) > p(1), and there is
an integer t such that
p(t) >
and
p(t − 1) + p(t + 1)
2
(1)
p(s)
p(y)
≥
,
s+1
y+1
for every 0 ≤ s ≤ y and y ∈ [t−1, t, t+1]. Then, determining
whether q(G, k, p) ≤ σ for a given graph G, integer k, and
threshold σ is an NP-complete problem.
While the conditions in Proposition 8 seem overly complicated, they are quite easy to satisfy. Assume that we are given
a penalty function that is concave in [−1, ∞], and p(−1) = 0.
Then due to concavity we have
p(x + 1)
p(x)
≥
, for
x+1
x+2
This leads to the following corollary.
x≥0
.
Corollary 9. Assume a monotonic penalty function p : Z → R
such that p(x) = 0 for x < 0, p(2) > p(1), and p is concave
and non-linear in [−1, s] for some s ≥ 1. Then, determining
whether q(G, k, p) ≤ σ for a given graph G, integer k, and
threshold σ is NP-complete problem.
Note that, due to proper inequality in Equation 1, p must
be non-linear. This condition is needed since pl satisfies every
other requirement.√Corollary 9 covers many penalty functions
such as p(x) = x + 1 or p(x) = log(x + 2), for x ≥ 0.
Note that function needs to be convex only in [−1, s] for some
s ≥ 1. At extreme, s = 1 in which case t = 0 satisfies the
conditions in Proposition 8.
V. R ELATED WORK
The problem of ranking an object based on its dominating
relationships to other objects is a classic problem. Perhaps the
most known ranking method is Elo rating devised by Elo [3],
used to rank chess players. In similar fashion, Jameson et al.
[8] introduced a statistical model, where the likelihood of the
the vertex dominating other is based on the difference of their
ranks, to animal dominance data.
Maiya and Berger-Wolf [11] suggested an approach for
discovering hierarchies, directed trees from weighted graphs
such that parents tend to dominate the children. To score
such a hierarchy they propose a statistical model where the
probability of an edge is high between a parent and a child.
To find a good hierarchy the authors employ a greedy heuristic.
The technical relationship between our approach and the
previous studies on agony by Gupte et al. [6] and Tatti [17] is
a very natural one. The authors of both papers demonstrate
that minimizing agony in a unweighted graph is a dual
problem to finding a maximal eulerian subgraph, a subgraph
in which, for each vertex, the same number of outdoing edges
and the number of incoming edges is the same. Discovering
the maximum eulerian subgraph is a special case of the
capacitated circulation problem, where the capacities are set to
1. However, the algorithms in [6, 17] are specifically designed
to work with unweighted edges. Consequently, if our input
graph has weighed edges or we wish to enforce the cardinality
constraint, we need to use the capacitated circulation solver.
The stark difference of computational complexities for
different edge penalties is intriguing: while we can compute agony and any other convex score in polynomial-time,
minimizing the concave penalties is NP-hard. Minimizing
the score q(G, k, pc ) is equivalent to FEEDBACK ARC SET
(FAS), which is known to be APX-hard with a coefficient
of c = 1.3606 [1]. Moreover, there is no known constantratio approximation algorithm for FAS, and the best known
approximation algorithm has ratio O(log n log log n) [4]. In
this paper we have shown that minimizing concave penalty is
NP-hard. An interesting theoretical question is whether this
optimization problem is also APX-hard, and is it possible to
develop an approximation algorithm.
Role mining, where vertices are assigned different roles
based on their adjacent edges, and other features, has received
some attention. Henderson et al. [7] studied assigning roles
to vertices based on its features while McCallum et al. [12]
assigned topic distributions to individual vertices. A potential
direction for a future work is to study whether we can use the
obtained ranking as a feature in role discovery.
Table I
BASIC CHARACTERISTICS OF THE DATASETS AND THE EXPERIMENTS . T HE
6 TH COLUMN IS THE NUMBER OF GROUPS IN THE OPTIMAL RANKING .
T HE COLUMNS NAMED iter. REPRESENT THE NUMBER OF ITERATIONS
NEEDED BY THE CAPACITATED CIRCULATION SOLVER .
VI. E XPERIMENTS
2 The
source code is available at http://research.ics.aalto.fi/dmg/
max SCC
403k
63k
265k
76k
82k
876k
7k
32
258
37k
425k
303k
|E| |V
with SCC
|E 0 |
|r| iter.
w/o SCC
time iter.
time
base
3m 395k 3m 17 9k 6h24m 3k 7h29m 4h27m
148k 14k 51k 24 83
8s 136
45s
45s
419k 34k 151k 9 1k
10m 8k
3h8m
2m
509k 32k 444k 10 4k
49m 4k
1h5m
20m
870k 71k 841k 9 4k 1h38m 4k 2h21m 1h5m
5m 435k 3m 31 52k 8h50m 6k 26h43m 2h32m
104k
1k 39k 12 738
43s 1k
5m
7s
205
32 205 6 157 22ms 157
22ms
–
4232
1
0 19
0 10ms 454
1.2s
–
31k 263 569 11 3k
0.2s 7k
10m
–
734k 13k 64k 22 5k
10m 45k 31h34m
–
445k
5k 20k 21 12k
2m 45k 19h38m
–
×104
4
2
0
5 10 15 20
constraint k
number of iterations
Amazon
Gnutella
EmailEU
Epinions
Slashdot
Google
WikiVote
Nfl
Reef
HiggsRep.
HiggsRet.
HiggsM.
0|
time (in 103 sec)
|V |
Name
q(G, k)
In this section we present our experimental evaluation.
Datasets and setup For our experiments we took 10 large
networks from SNAP repository [9]. In addition, for illustrative
purposes, we used two small datasets: Nfl, consisting of
National Football League teams. We created an edge (x, y) if
team x has scored more points against team y during 2014
regular season, we assign the weight to be the difference
between the points. Since not every team plays against every
team, the graph is not a tournament. Reef, a food web of
guilds of species [15], available at [16]. The dataset consisted
of 3 food webs of coral reef systems: The Cayman Islands,
Jaimaica, and Cuba. An edge (x, y) appears if a guild x is
known to prey on a guild y. Since the guilds are common
among all 3 graphs, we combined the food webs into one
graph, and weighted the edges accordingly, that is, each
edge received a weight between 1 and 3. The sizes of the
graphs, along with the sizes of the largest strongly connected
component, are given in the first 4 columns of Table I.
The 3 Higgs and Nfl graphs had weighted edges, and for the
remaining graphs we assigned a weight of 1 for each edge. We
removed any self-loops as they have no effect on the ranking,
as well as any singleton vertices.
For each dataset we computed the agony. To solve the
C IRCULATION problem we used algorithm by Edmonds and
Karp [2]. While the algorithm by Orlin [13] has better theoretical bounds, the pathological case that is solved by Orlin
[13] does not occur in practice for our datasets. For the
unweighted graphs we also computed the agony using R ELIEF,
an algorithm suggested by Tatti [17]. Note that this algorithm,
nor the algorithm by Gupte et al. [6], does not work for
weighted graphs nor when the cardinality constraint k given in
Problem 1 is enforced. We implement both algorithms in C++
and performed experiments using a Linux-desktop equipped
with a Opteron 2220 SE processor.2
Results We begin with running times given in Table I. We
report the running times of our approach with and without the
strongly connected component decomposition as suggested by
Proposition 5, and compare it against the baseline R ELIEF,
whenever possible. Note that we can use the decomposition
only if we do not enforce the cardinality constraint.
Our first observation is that the decomposition always helps
to speed up the algorithm. In fact, this speed-up may be
dramatic, if the size of the strongly connected component
is significantly smaller than the size of the input graph, for
example, with HiggsRetweet. The number of iterations of iterations needed to solve the circulation problem is significantly
smaller than the worst case scenario of O(m log n), making
this algorithm feasible even for large graphs. Interestingly
enough, this number increases in some cases, when using
6
4
2
0
5 10 15 20
constraint k
×104
2
1.5
1
0.5
0
5 10 15 20
constraint k
Figure 4. Agony, execution time, and number of iterations needed by
capacitated circulation solver as a function of the constraint k for Gnutella.
the decomposition, but a single iteration may be significantly
faster when the decomposition is used.
The baseline R ELIEF is almost always faster than the approach, even though the computational complexity guarantees,
O(m2 ) for R ELIEF and O(m2 log n + mn log2 n) for CANON,
are practically the same. Nevertheless, the difference between
the computation times is less than the order of magnitude.
Our next step is to study the effect of the constraint k, the
maximum number of different rank values. We see in the 6th
column in Table I that despite having large number of vertices,
that the optimal rank assignment has low number of groups,
typically around 10–20 groups, even if the cardinality constraint is not enforced. This suggests the following strategy to
compute the agony when the cardinality constraint is enforced
and the input graph has small connected components: First
compute the agony quickly without the constraint using the
strongly connected component decomposition. If the obtained
ranking has less than k components, then we are done.
Otherwise, enforce the constraint k and solve the problem
without the decomposition.
Next, we consider agony as a function of k, which we have
plotted in Figure 4 for Gnutella. We see that agony remains
relatively constant as we decrease k, and starts to increase
more prominently once we consider assignments with k ≤ 5.
Enforcing the constraint k has an impact on running time.
In Figure 4 we plotted the running time as a function of k.
As we can see lower values of k are computationally more
taxing to solve even though they have the same theoretical
bounds on running time. The main reason for more additional
computational burden is that solving agony with small values
of k requires more iterations.
Table II
R ANK ASSIGNMENT DISCOVERED FOR Nfl DATASET WITH k = 3 GROUPS
Rank
Teams
1.
2.
3.
DEN BAL NE DAL SEA PHI KC GB PIT
STL NYG MIA CAR NO SD MIN CIN BUF DET IND HOU SF ARI
WSH OAK TB JAX TEN CLE ATL NYJ CHI
Let us look on the ranking that we obtained from Nfl dataset
using k = 3 groups, given in Table II. The obtained ranking
is very sensible: 7 of 8 teams in the top group consists of
playoff teams of 2014 season, while the bottom group consists
of teams that have a significant losing record.
Finally, let us look on rankings obtained Reef dataset. The
graph is in fact a DAG with 19 groups. To reduce the number
of groups we rank the guilds into k = 4 groups. The condensed
results are given in Table III. We see that the top group consists
of large fishes and sharks, the second group contains mostly
smaller fishes, a large portion of the third group are crustacea,
while the last group contains the bottom of the food chain,
planktons and algae. We should point out that this ranking
is done purely on food web, and not on type of species. For
example, cleaner crustacea is obviously very different than
plankton. Yet cleaner crustacea only eats planktonic bacteria
and micro-detritivores while being eaten by many other guilds.
Consequently, it is ranked in the bottom group.
VII. C ONCLUDING REMARKS
In this paper we studied the problem of discovering a
hierarchy in a directed graph that minimizes agony. We
introduced several natural extensions: (i) we demonstrated how
to compute the agony for weighted edges, and (ii) how to
limit the number of groups in a hierarchy. Both extensions
cannot be handled with current algorithms, hence we provide
a new technique by demonstrating that minimizing agony can
be solved by solving a capacitated circulation problem, a wellknown graph problem with a polynomial solution.
We can further generalize the setup by allowing each edge
to have its own individual penalty function. As long as the
penalty functions are convex, the construction done in Section IV-A can be used to solve the optimization problem. We
can further generalize the cardinality constraint by requiring
that only a subset of vertices must have ranks within some
range. We can have multiple such constraints.
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